This is a lesson which requires diagrams to help students model the problems. Visualizing and modeling help students make sense of what is being asked of them in the problem. (mp1) Lessons and activities that require student perseverence may be difficult for a substitute teacher to manage, because he/she may think that if students appear to be struggling that the math is too hard for them and the lesson isn't working. I like to set the teacher up with some questions and suggestions that they can make without jumping in to help too quickly or giving away the answers.

Students will begin working on a problem set that will take a couple of days. They will need to reconvene in tomorrow's lesson to debrief and look for patterns. This lesson is designed to show students the benefit of drawing a diagram to model the problem and also to generate some excitement about the design aspects in our apartment unit.

This warm up says that two girls, Karla & Nathalie, both have rectangular rooms.

Karla says that her room has a larger area than Nathalie's. Nathalie says that her room has a larger perimeter. Students are asked what the dimensions of the two rooms might be.

Because students often show resistance to this type of problem where they are not told exactly what to do I ask the substitute to do what I often do which is to circulate and encourage students to try out some dimensions and keep adjusting them until they come upon some that fit the criteria. I also suggest that they try a diagram.

If students are trying unsuccessfully and begin to think it isn't possible I suggest that the substitute give them the example in which Karla's room is 5 ft. by 6 ft., giving an area of 30 sq. ft. and a perimeter of 25 ft. and Nathalie's room is 2 ft. by 11 ft, giving a smaller area of 22 sq. ft. and a larger perimeter of 26 ft. Once students see it is possible they are more likely to try to come up with other possibilities. I really just want them to practice calculating area and perimeter and distinguishing one from the other. They get practice whether or not they actually come up with any examples that fit the criteria.

When students have had some time to generate a few examples the sub is asked to have students share their solutions and have other students test that they fit the criteria. By testing they are learning not only to critique the claims of others, but are also getting additional practice calculating area and perimeter.

Resources

Students are given their border problem which is a set of "border" problems. They are told that Samantha is designing a flower garden. She wants the garden to be a square measuring 10 feet on each side. She wants to make a border out of tiles around the outside perimeter of her garden. The tiles she wants are 1 foot by 1 foot squares. Students must figure out how many tiles Samantha will need without counting the tiles individually. They are asked to show multiple ways and show the specific math they have to do. The initial problem that introduces the idea has a visual with a white square bordered by a darker band. To begin I ask the sub to have students explain the figure with words to help them make sense of the problem. Students need to be able to visualize that the 10 by 10 garden is contained within the white square and the darker band represents where the tiles are for the border. The sub may use the border problem visual on the projector to help students make sense of the problem before they begin. I also provide the sub with border problem visual scaffold notes to help guide the discussion.

The classwork/homework asks students to solve similar problems using squares of different dimensions. The last two questions are extension questions. One asks students to write an expression to show how many tiles would be needed to border a square of any dimensions and the last question asks students how many tiles are needed if the border is being built inside the square instead of outside. I ask the sub to encourage the students to draw diagrams to model the problem in order to help them make sense of them.